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As a recap, I have a DIY canister (see photos) of nearly 4 liters of a Matrix/De*Nitrate mix. I'm using a Tom aqua lifter pump which was delivering 3.5gph. I used Stability to bio-seed the filter even though my 60g tank was well established. I have replaced media in my Aquaclear 70 filter a few times so that it is mechanical only (forcing biological filtration in the canister).
I subsequently added the sponge canister and have most recently reduced the flow to a trickle.
(note: Water flows from the tank, through the matrix/de*nitrate canister, then through the sponge canister and back to the tank).
Although ammonia and nitrites appear to be handled from the beginning, I have yet to see any evidence of nitrate reduction.

Is there anything else I might do to encourage development of anaerobic bacteria in the matrix/de*nitrate to handle nitrates?

Out of curiosity, would you be able to have water flow first through your mechanical filter (sponge), than through the Matrix/de*nitrate? We typically recommend having filtration media in the following order, but realize this is not always an option:

mechanical --> chemical --> biological

Having the water flow through the sponge first will allow for the bigger, undissolved particulates to be removed from the water before reaching the chemical and biological, which will help them to work more efficiently and not become clogged.

Now that you have dosed the Stability for a week, you may want to put the Purigen back into the filter to help with nitrogenous organic waste.

Since your source water already has such a high level of nitrates, plus whatever waste that your inhabitants are producing, the bacteria are having to work extra hard in order to keep them at a manageable levels. You have plenty of media in your filter, however, if with routine water changes, and frequent Stability dosing, you are finding that your nitrate levels are remaining at a higher level (40-60ppm), the next step might be to change your source water. Or at least dilute it with RO/DI water, which will lower the incoming nitrate levels so that the beneficial bacteria can work more easily.

Lets back up just a bit. I have an Aquaclear 70 HOB filter to handle mechanical/chemical filtration. The DIY bio-denitrate filter is dedicated for bio. The siphon inlet tube is high in the tank and will see little/no detritus.
Originally, the matrix/de*nitrate canister was alone (without the sponge canister). After seeing no progress in weeks, I added the sponge filled canister. I setup the sponge/floss filter after the matrix/de*nitrate so the bacteria in the first canister would have the benefit of any dissolved organics (food). The sponge/floss would then polish the water before returning to the tank.
So I could switch them around, but all things considered, I think my logic holds.

I used Fluval Nitrate Remover (FNR) to lower the tank nitrates to 5-10ppm. I am using an API Tap Water filter to make DI water. I setup a spare 10g tank in the garage to filter well water with FNR to get water with nitrates 5-10ppm. I mix 5g of treated DI water with 5g of filtered well water for a 10g weekly water change (60g tank). This has been in play since the filter was setup. This water is <=5ppm nitrates.

I have had Purigen in the AC70. I also have several bunches of anacharis floating plants. The nitrates in the tank just slowly rise. When they got to 20/40ppm, I used the FNR in the AC70 tank filter again to bring the nitrates down.

The effluent from the DIY filter shows that nitrate levels are exactly the same as tank water.
I used up the initial 250ml of Stability and have recently finished another 250ml.
It just doesn't seem to be working which it why I just reduced the flow to further starve the chambers of oxygen hoping for anaerobic bacteria to develop.

Do you have a fairly heavily-stocked tank? When it comes to controlling nitrates, the key is to remove them at a greater rate than they are being produced. It sounds like the nitrates are being produced at a greater rate than they are being removed at this point. Reducing the flow to your DIY chamber should allow more denitrifying bacteria to colonize the de*nitrate, as it does require a fairly slow flow rate. The Matrix, on the other hand, should already have some good colonies established.

I have 4 young adult Platy's, 10 Platy fry (1/4" or so) and 6 neon tetras (again, 60g tank).
I think the plants and Purigen is preventing some nitrates, but so far in spite of the volume of product and amount of Stability, although ammonia and nitrites seem fine, there is no evidence that the DIY matrix/de*nitrate filter is processing any nitrates at all....at least not yet. The Tom Aqua Lifter only pumps at 3.5gph which is very low. Reducing to just a trickle (a bit more than drops) to starve O2 is kind of a hail mary to try and get it working.
(slowing to a drip rate, I should be able to have anaerobic bacteria development even in sponge material.)

I think at this point it is a matter of allowing your biofilter to mature. The bacteria that were added through Stability will continue to proliferate, eventually being able to keep up with the demands of the tank.

On a related note, a few weeks in with the DIY filter I had a fitting leak on the Matrix/De*Nitrate canister. When I emptied the canister into a large bowl, I discovered a fair amount of bio-slime on the product. When I reassembled the filter, I rinsed the product (well water - no chlorine) to remove.
It would seem that bacteria produced bio-slime might be counter productive to how Matrix/De*Nitrate functions as it would/could block pores resulting in reduced efficiency.
Am I correct in this? Should the media be periodically rinsed to remove bio slime? If so, are there any guidelines or your opinion on frequency of cleaning?